Question

The binomial and Poisson distributions are two different discrete probability distributions. Explain the differences between the distributions and provide an example of how they could be used in the healthcare industry.

Identify the functions for binomial, Poisson, and normal distributions and discuss how Excel can be used to calculate probabilities of X, <X, and >X. Apply an example to at least one business scenario.

Answer 1

A probability distribution is defined as a mathematical function that provides the probabilities of the occurrences of several different possible outcomes in a statistical experiment. There are two types of probability distributions i.e, discrete or continuous. In continuous probability distributions, the function describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable that has a set of possible values (known as the range) that is infinite and uncountable. A discrete probability distribution tries to describe the probability of a discrete random variable. A discrete random variable is a variable that has countable values such as a list of non-negative integers.

Key Differences between them are:

Meaning: Binomial Distribution can be applied to those statistical experiments where one has to study the probability of the repeated number of trials while the Poisson's Distribution actually tries to return the count of independent events that occur randomly within a given period of time.

Outcomes: The Binomial Distribution has only two possible outcomes while the Poisson's Distribution has an infinite number of possible outcomes.

Number of trials: The Binomial Distribution has a fixed number of trials while the Poisson's Distribution has an infinite number of trials.

Mean and Variance: In the case of Binomial Distribution the Mean > Variance while in the case of Poisson's Distribution the Mean = Variance.

Success: Binomial Distribution yields a constant probability while the Poisson's Distribution may even yield an infinitesimal chance of success.

Parametric Nature: Binomial Distribution is biparametric while the Poisson's Distribution is uniparametric.

Example: Probability Distributions are widely used by researchers to perform experiments in Medical Field. For example, if a researcher is working on a problem of determining whether the patient will die or survive. The researcher is actually concerned about the occurrence of an event and not in its magnitude. So the researcher basically uses the Binomial Distribution to find out out how many patients survive given their medical condition. In another example, a researcher may want to use Poisson's Distribution to find the correct number of deaths that take place in a town due to a particular disease every day. In the last case, the Poisson's Distribution will be a better choice as the time period is specified.

To implement the Binomial Distribution and Poisson's Distribution in Excel follow the below steps:

Step1: Open MS Excel

Step2: Load/Prepare the Data

Step3: Go to the formula tab and select more functions tab.

Step 4: From the more functions tab select the Statistical Functions.

Step 5: From the dropdown select BINOM.DIST/POISSON.DIST.

Step 6: Apply the distributions on the data and represent them suitably.

For the BINOM.DIST y9ou are supposed to input the total number of successes, the total number of trials, the probability of the success, specify True to use the cumulative distribution function and False to use the probability mass function.

Here's the solution to your problem. Thanks for asking and happy learning!!